Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

38,383 questions
9k views

3k views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
64k views

What is the math behind the game Spot It?

I just purchased the game Spot It. As per this site, the structure of the game is as follows: Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 ...
7k views

Can $18$ consecutive integers be separated into two groups,such that their product is equal?

Can $18$ consecutive positive integers be separated into two groups, such that their product is equal? We cannot leave out any number and neither we can take any number more than once. My work: ...
35k views

Probability that random moves in the game 2048 will win

I have recently played the game 2048, created by Gabriele Cirulli, which is fun. I suggest trying if you have not. But my brother posed this question to me about the game: If he were to write a ...
6k views

Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into ...
3k views

In combinatorics, how can one verify that one has counted correctly?

This is a soft question, but I've tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly. It's ...
47k views

33k views

A comprehensive list of binomial identities?

Is there a comprehensive resource listing binomial identities? I am more interested in combinatorial proofs of such identities, but even a list without proofs will do.
3k views

Choosing office hours to maximise number of students who can attend at least one time slot

I have the following (real world!) problem which is most easily described using an example. I ask my six students when the best time to hold office hours would be. I give them four options, and say ...
2k views

Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? ...
Guessing a subset of $\{1,…,N\}$
I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need?